具有奇异相互作用核的动力学McKean-Vlasov SDEs的混沌与涨落的定量传播
Quantitative Propagation of Chaos and Fluctuations for Kinetic McKean--Vlasov SDEs with Singular Interaction Kernels
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中文总结 AI 辅助
研究一类含奇异相互作用核的退化动力学McKean-Vlasov SDEs相关粒子系统，通过动力学Krylov-Khasminskii估计等方法，证明混沌定量传播估计和中心极限定理，得到路径空间相对熵界及Berry-Esseen型界。
中文摘要 AI 辅助
我们证明了一类具有外部漂移和Kato类奇异相互作用核的退化动力学McKean-Vlasov SDEs相关粒子系统的混沌定量传播估计和中心极限定理。特别地，相互作用核可在混合\(L^q_tL^{p_v}_vL^{p_x}_x\)空间，其中\(\frac2q+\frac{3d}{p_x}+\frac d{p_v}<1\)。对于相关\(N\)粒子系统，在仅假设初始数据的熵混沌性下，得到前\(k\)个粒子的路径空间相对熵界为\(k/N\)阶。关键要素是动力学Krylov-Khasminskii估计和经验相互作用场的条件希尔伯特空间次高斯估计。对于CLT，还证明了有限维投影的Berry-Esseen型界。
英文摘要
We prove a quantitative propagation of chaos estimate and a central limit theorem for the particle system associated with a class of degenerate kinetic McKean--Vlasov SDEs with external drifts and singular interaction kernels in Kato's class. In particular, the interaction kernel can be in the mixed $L^q_tL^{p_v}_vL^{p_x}_x$-space, where $\frac2q+\frac{3d}{p_x}+\frac d{p_v}<1$. For the associated $N$-particle system, we obtain a path-space relative entropy bound of order $k/N$ for the first $k$ particles, assuming only entropic chaoticity of the initial data. The key ingredients are kinetic Krylov--Khasminskii estimates and a conditional Hilbert-space subgaussian estimate for empirical interaction fields. For the CLT, we also prove a Berry--Esseen-type bound for finite-dimensional projections.
